Search

Under high perpendicular magnetic field and at low temperatures, graphene develops an insulating state at the charge neutrality point. This state, dubbed $\nu=0$, is due to the interplay between electronic interactions and the four-fold spin and valley degeneracies in the flat band formed by the $n=0$ Landau level. Determining the ground state of $\nu=0$, including its spin and valley polarization, has been a theoretical and experimental undertaking for almost two decades. Here, we present experiments probing the bulk thermal transport properties of monolayer graphene at $\nu=0$, which directly probe its ground state and collective excitations. We observe a vanishing bulk thermal transport, in contradiction with the expected ground state, predicted to have a finite thermal conductance even at very low temperature. Our result highlight the need for further investigations on the nature of $\nu=0$., Comment: Contains Supplementary Information

In the first part of this paper, we develop a general framework that permits a comparison between explicit class field theories for a family of rational function fields $\mathbb{F}_s(t)$ over arbitrary constant fields $\mathbb{F}_s$ and explicit class field theory for the rational function field $\mathfrak{K}(t)$ over the nonprincipal ultraproduct $\mathfrak{K}$ of the constant fields $\mathbb{F}_s$. Under an additional assumption that the constant fields $\mathbb{F}_s$ are perfect procyclic fields, we prove a correspondence between ramifications of primes $P$ in $\mathfrak{K}(t)$ and ramifications of primes $P_s$ in $\mathbb{F}_s(t)$, where the $P_s$ are primes in $\mathbb{F}_s(t)$ whose nonprincipal ultraproduct coincides with $P$. In the second part of the paper, we are mainly concerned with rational function fields over a large class of fields, called $n$-th level ultra-finite fields that are a generalization of finite fields. At the $0$-th level, ultra-finite fields are simply finite fields, and for an arbitrary positive integer $n$, an $n$-th level ultra-finite field is inductively defined as a nonprincipal ultraproduct of $(n - 1)$-th level ultra-finite fields. We develop an analogue of cyclotomic function fields for rational function fields over $n$-th level ultra-finite fields that generalize the works of Carlitz and Hayes for rational function fields over finite fields such that these cyclotomic function fields are in complete analogy with the classical cyclotomic fields $\mathbb{Q}(\zeta)$ of the rationals $\mathbb{Q}$. The main result in the second part of the paper is an analogue of the Kronecker--Weber theorem for rational function fields over $n$-th level ultra-finite fields that explicitly describes, from a model-theoretic viewpoint, the maximal abelian extension of the rational function field over a given $n$-th level ultra-finite field for all $n \ge 1$.

In this paper, we prove the existence of a first-order definition of the polynomial ring over a nonprincipal ultraproduct of finite fields of unbounded cardinalities in its fraction field by a universal-existential formula in the language of rings augmented by an additional constant symbol $t$. As a consequence, we prove that the full first-order theory of the rational function field over a nonprincipal ultraproduct of finite fields of characteristic $0$ is undecidable., Comment: Added references and a new result regarding undecidability

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring $\mathbb{F}_q[t]$ over a finite field $\mathbb{F}_q$ that is due to Dedekind, K\"uhne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics $0$ and $p >0$ for some prime $p$. We then use the higher reciprocity law to prove an analogue of the Grunwald--Wang theorem for such a polynomial ring in both characteristics $0$ and $p > 0$ for some prime $p$., Comment: We revised the first version of the paper to include any nonprincipal ultraproduct of finite fields of both characteristics $0$ and $p >0$. The ultraproduct can be taken over the same finite field $\mathbb{F}_q$, in which case the ultraproduct becomes the finite field $\mathbb{F}_q$, which allows to recover the classical reciprocity law for $\mathbb{F}_q[t]$

In superconductor-normal-superconductor (SNS) junctions, dissipationless supercurrent is mediated via Andreev bound states (ABSs) controlled by the phase difference between the two superconductors. Theory has long predicted significant fluctuations, and thus a noise, of such supercurrent in equilibrium, due to the thermal excitation between the ABSs. Via the fluctuation-dissipation theorem (FDT), this leads paradoxically to a dissipative conductance even in the zero frequency limit. In this article, we directly measure the supercurrent noise in a phase-biased SNS ring inductively coupled to a superconducting resonator. Using the same setup, we also measure its admittance and quantitatively demonstrate the FDT for any phase. The dissipative conductance shows an 1/T temperature dependence, in contrast to the Drude conductance of unproximitized metal. This is true even at phase $\pi$ where the Andreev spectrum is gapless. Supported by linear response theory, we attribute this to the enhanced current correlation between the ABSs symmetrical across the Fermi level. Our results provide insight into the origin of noise in hydrid superconducting devices which may be useful for probing topologically protected ABSs.

Universal methods for optimization are designed to achieve theoretically optimal convergence rates without any prior knowledge of the problem's regularity parameters or the accurarcy of the gradient oracle employed by the optimizer. In this regard, existing state-of-the-art algorithms achieve an $\mathcal{O}(1/T^2)$ value convergence rate in Lipschitz smooth problems with a perfect gradient oracle, and an $\mathcal{O}(1/\sqrt{T})$ convergence rate when the underlying problem is non-smooth and/or the gradient oracle is stochastic. On the downside, these methods do not take into account the problem's dimensionality, and this can have a catastrophic impact on the achieved convergence rate, in both theory and practice. Our paper aims to bridge this gap by providing a scalable universal gradient method - dubbed UnderGrad - whose oracle complexity is almost dimension-free in problems with a favorable geometry (like the simplex, linearly constrained semidefinite programs and combinatorial bandits), while retaining the order-optimal dependence on $T$ described above. These "best-of-both-worlds" results are achieved via a primal-dual update scheme inspired by the dual exploration method for variational inequalities., Comment: 31 pages, 4 figures, 1 table; to appear in ICML 2022

We consider the traffic assignment problem in nonatomic routing games where the players' cost functions may be subject to random fluctuations (e.g., weather disturbances, perturbations in the underlying network, etc.). We tackle this problem from the viewpoint of a control interface that makes routing recommendations based solely on observed costs and without any further knowledge of the system's governing dynamics -- such as the network's cost functions, the distribution of any random events affecting the network, etc. In this online setting, learning methods based on the popular exponential weights algorithm converge to equilibrium at an $\mathcal{O}({1/\sqrt{T}})$ rate: this rate is known to be order-optimal in stochastic networks, but it is otherwise suboptimal in static networks. In the latter case, it is possible to achieve an $\mathcal{O}({1/T^{2}})$ equilibrium convergence rate via the use of finely tuned accelerated algorithms; on the other hand, these accelerated algorithms fail to converge altogether in the presence of persistent randomness, so it is not clear how to achieve the "best of both worlds" in terms of convergence speed. Our paper seeks to fill this gap by proposing an adaptive routing algortihm with the following desirable properties: $(i)$ it seamlessly interpolates between the $\mathcal{O}({1/T^{2}})$ and $\mathcal{O}({1/\sqrt{T}})$ rates for static and stochastic environments respectively; $(ii)$ its convergence speed is polylogarithmic in the number of paths in the network; ${(iii)}$ the method's per-iteration complexity and memory requirements are both linear in the number of nodes and edges in the network; and ${(iv)}$ it does not require any prior knowledge of the problem's parameters.

In their recent work, Gentili and Struppa proposed a different quaternionic analogue of the notion of holomorphic functions in the complex plane, called \textit{slice regular functions}, which has led to several analogues of classical theorems in complex function theory in the quaternionic setting. The quaternionic analogue of meromorphic functions is called \textit{semiregular functions}. Such a function is said to be \textit{slice preserving} if it maps each domain in a complex line to a domain in the same complex line. In this work, we deal with the relation between analytic and algebraic properties of certain fields of semiregular functions in symmetric slice domains of the real quaternions. More precisely, we prove quaternionic analogues of Bers's theorem and Iss'sa's theorem that deal with the algebraic structures of the ring of slice preserving regular functions and of certain fields of slice preserving semiregular functions on symmetric slice domains in the real quaternions.

We introduce the Colonel Blotto game with favoritism, an extension of the famous Colonel Blotto game where the winner-determination rule is generalized to include pre-allocations and asymmetry of the players' resources effectiveness on each battlefield. Such favoritism is found in many classical applications of the Colonel Blotto game. We focus on the Nash equilibrium. First, we consider the closely related model of all-pay auctions with favoritism and completely characterize its equilibrium. Based on this result, we prove the existence of a set of optimal univariate distributions -- which serve as candidate marginals for an equilibrium -- of the Colonel Blotto game with favoritism and show an explicit construction thereof. In several particular cases, this directly leads to an equilibrium of the Colonel Blotto game with favoritism. In other cases, we use these optimal univariate distributions to derive an approximate equilibrium with well-controlled approximation error. Finally, we propose an algorithm -- based on the notion of winding number in parametric curves -- to efficiently compute an approximation of the proposed optimal univariate distributions with arbitrarily small error., Comment: Full paper version of ACM EC conference paper: https://doi.org/10.1145/3465456.3467527

This paper presents a cloud-connected indoor air quality sensor system that can be deployed to patients' homes to study personal microenvironmental exposure for asthma research and management. The system consists of multiple compact sensor units that can measure residential NO2, ozone, humidity, and temperature at one minute resolution and a cloud based informatic system that acquires, stores, and visualizes the microenvironmental data in real time. The sensor hardware can measure NO2 as low as 10 ppb and ozone at 15 ppb. The cloud informatic system is implemented using open-source software on Amazon Web Service for easy deployment and scalability. This system was successfully deployed to pediatric asthma patients' homes in a pilot study. In this study, we discovered that some families can have short term NO2 exposure higher than EPA's one hour exposure limit (100 ppb), and NO2 micropollution episodes often arise from natural gas appliance usage such as gas stove burning during cooking. By combining the personalized air pollutant exposure measurements with the physiological responses from a patient diary and medical record, this system can enable novel asthma research and personalized asthma management.

Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One of the main contributions of our paper is to introduce a new topological structure to hypergraph data which bears a resemblance to a usual metric space structure. Using this new topological space structure of hypergraph data, we propose several approaches to study community detection problem, detecting persistent features arising from homological structure of hypergraph data. Also based on the topological space structure of hypergraph data introduced in our paper, we introduce a modified nearest neighbors methods which is a generalization of the classical nearest neighbors methods from machine learning. Our modified nearest neighbors methods have an advantage of being very flexible and applicable even for discrete structures as in hypergraphs. We then apply our modified nearest neighbors methods to study sign prediction problem in hypegraph data constructed using our method.

A weighted directed network (WDN) is a directed graph in which each edge is associated to a unique value called weight. These networks are very suitable for modeling real-world social networks in which there is an assessment of one vertex toward other vertices. One of the main problems studied in this paper is prediction of edge weights in such networks. We introduce, for the first time, a metric geometry approach to studying edge weight prediction in WDNs. We modify a usual notion of WDNs, and introduce a new type of WDNs which we coin the term \textit{almost-weighted directed networks} (AWDNs). AWDNs can capture the weight information of a network from a given training set. We then construct a class of metrics (or distances) for AWDNs which equips such networks with a metric space structure. Using the metric geometry structure of AWDNs, we propose modified $k$ nearest neighbors (kNN) methods and modified support-vector machine (SVM) methods which will then be used to predict edge weights in AWDNs. In many real-world datasets, in addition to edge weights, one can also associate weights to vertices which capture information of vertices; association of weights to vertices especially plays an important role in graph embedding problems. Adopting a similar approach, we introduce two new types of directed networks in which weights are associated to either a subset of origin vertices or a subset of terminal vertices . We, for the first time, construct novel classes of metrics on such networks, and based on these new metrics propose modified $k$NN and SVM methods for predicting weights of origins and terminals in these networks. We provide experimental results on several real-world datasets, using our geometric methodologies.

In many real world datasets arising from social networks, there are hidden higher order relations among data points which cannot be captured using graph modeling. It is natural to use a more general notion of hypergraphs to model such social networks. In this paper, we introduce a new local geometry of hyperdges in hypergraphs which allows to capture higher order relations among data points. Furthermore based on this new geometry, we also introduce new methodology--the nearest neighbors method in hypergraphs--for analyzing datasets arising from sociology.

For a discrete valuation domain $V$ with maximal ideal $\mathfrak{m}$ such that the residue field $V/\mathfrak{m}$ is finite, there exists a sequence of polynomials $(F_n(x))_{n \ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V) = \{f \in K[x] | f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials $(\binom{x}{n})_{n \ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfrak{m}$, and prove a result that provides a method for counting exactly how many $V$-values of the polynomials $(F_n(x))_{n \ge 0}$ fall into each of the residue classes modulo $\mathfrak{m}$. Our main theorem in this paper can be viewed as an analogue of the classical theorem of Garfield and Wilf in the context of discrete valuation domains.

Let $k = \mathbb{Q}(\sqrt{\alpha})$ be a real quadratic number field, where $\alpha$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ instead of bounded generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ as considered in this paper.

Resource allocation games such as the famous Colonel Blotto (CB) and Hide-and-Seek (HS) games are often used to model a large variety of practical problems, but only in their one-shot versions. Indeed, due to their extremely large strategy space, it remains an open question how one can efficiently learn in these games. In this work, we show that the online CB and HS games can be cast as path planning problems with side-observations (SOPPP): at each stage, a learner chooses a path on a directed acyclic graph and suffers the sum of losses that are adversarially assigned to the corresponding edges; and she then receives semi-bandit feedback with side-observations (i.e., she observes the losses on the chosen edges plus some others). We propose a novel algorithm, EXP3-OE, the first-of-its-kind with guaranteed efficient running time for SOPPP without requiring any auxiliary oracle. We provide an expected-regret bound of EXP3-OE in SOPPP matching the order of the best benchmark in the literature. Moreover, we introduce additional assumptions on the observability model under which we can further improve the regret bounds of EXP3-OE. We illustrate the benefit of using EXP3-OE in SOPPP by applying it to the online CB and HS games., Comment: This work was as replacement for arXiv:1905.11151 but was mistakenly submitted here as a new article

Non-invasive continuous alcohol monitoring has potential applications in both population research and in clinical management of acute alcohol intoxication or chronic alcoholism. Current wearable monitors based on transdermal alcohol content (TAC) sensing are relatively bulky and have limited quantification accuracy. Here we describe the development of a discreet wearable transdermal alcohol (TAC) sensor in the form of a wristband or armband. This novel sensor can detect vapor-phase alcohol in perspiration from 0.09 ppm (equivalent to 0.09 mg/dL sweat alcohol concentration at 25 {\deg}C under Henry's Law equilibrium) to over 500 ppm at one-minute time resolution. The TAC sensor is powered by a 110 mAh lithium battery that lasts for over 7 days. In addition, the sensor can function as a medical "internet-of-things" (IoT) device by connecting to an Android smartphone gateway via Bluetooth Low Energy (BLE) and upload data to a cloud informatics system. Such wearable IoT sensors may enable large-scale alcohol-related research and personalized management. We also present evidence suggesting a hypothesis that perspiration rate is the dominant factor leading to TAC measurement variabilities, which may inform more reproducible and accurate TAC sensor designs in the future., Comment: 11 pages, 7 figures